Statistics: Probability Distributions, Binomial Distribution
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1. Determine whether each of the distributions given below represents a probability distribution. Justify your answer.
(A)
x
1
2
3
4
P(x)
0.25
5/12
1/3
1/6
(B)
x
3
6
8
P(x)
0.1
3/5
0.3
(C)
x
20
30
40
50
P(x)
0.2
-0.2
0.7
0.3
2. Consider a binomial distribution with 13 identical trials, and a probability of success of 0.4
i. Find the probability that x = 3 using the binomial tables
ii. Use the normal approximation to find the probability that x = 3. Show all work.
3. The diameters of oranges in a certain orchard are normally distributed with a mean of 7.25 inches and a standard deviation of 0.50 inches. Show all work.
(A) What percentage of the oranges in this orchard have diameters less than 6.8 inches?
(B) What percentage of the oranges in this orchard are larger than 7.10 inches?
(C) A random sample of 100 oranges is gathered and the mean diameter obtained was 7.10. If another sample of 100 is taken, what is the probability that its sample mean will be greater than 7.10 inches?
(D) Why is the z-score used in answering (A), (B), and (C)?
(E) Why is the formula for z used in (C) different from that used in (A) and (B)?
4. Assume that the population of heights of female college students is approximately normally distributed with mean m of 72 inches and standard deviation s of 3.75 inches. A random sample of 15 heights is obtained. Show all work.
(A) Find the proportion of female college students whose height is greater than 75 inches.
(B) Find the mean and standard error of the distribution
(C) Find
5. Answer the following questions regarding the normal, standard normal and binomial distributions.
(A) How does the standard normal distribution differ from the normal distribution?
(B) What are the advantages of using the standard normal distribution over the normal distribution?
(C) Why is the correction for continuity necessary when the normal distribution is used to approximate a binomial distribution?
6. Two cards are selected, one at a time from a standard deck of 52 cards. Let x represent the number of Jacks drawn in a set of 2 cards.
(A) If this experiment is completed without replacement, explain why x is not a binomial random variable.
(B) If this experiment is completed with replacement, explain why x is a binomial random variable.
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1. Determine whether each of the distributions given below represents a probability distribution. Justify your answer.
(A)
x
1
2
3
4
P(x)
0.25
5/12
1/3
1/6
It is not a probability distribution as sum of the probabilities is greater than 1.
(B)
x
3
6
8
P(x)
0.1
3/5
0.3
Yes, it is a probability distribution as sum of the probabilities is 1.
(C)
x
20
30
40
50
P(x)
0.2
-0.2
0.7
0.3
It is not a probability distribution as probability can't be negative.
2. Consider a binomial distribution with 13 identical trials, and a probability of success of 0.4
i. Find the probability that x = 3 using the binomial tables
Solution:
Here n = 13, p = 0.4, q = 0.6
P(x = 3) = C(13, 3)*(0.4^3)*(0.6^10) = 0.1107
ii. Use the normal approximation to find the probability that x = 3. Show all work.
Mean = np = 13*0.4 = 5.2
Standard deviation
Z1 = (2.5 - 5.2)/1.766 = -1.529
Z2 = (3.5 - 5.2)/1.766 = -0.963
= P(-1.529 < z < -0.963) ...
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